THE STUDY OF NON-LINEAR PHENOMENA BY THE ASYMPTOTIC ANALYSIS

这个项目的目的是通过使用渐近分析来数学地描述非线性现象。1）Narukawa和Shagai提出了一个与非线性弹性力学相关的数学模型。这是由一个退化的拟线性椭圆方程，其主要部分有不同的订单在0和无穷远。他们给出了这类拟线性方程的非线性特征值问题正解的全局分歧图，特别是多个正解的共存。这些结果是通过对弱解的正则性估计和对Ambrosetti，Brezis和Cerami在半线性情形下给出的结论的修正得到的。2）Ohnuma研究了一类奇异退化抛物方程，包括p-Laplace扩散方程和平均曲率流方程，证明了这类方程的比较原理。他还发现了拟线性退化椭圆型方程的一个强极大值原理。3）Murakami给出了高阶线性差分方程不动点渐近稳定的一个充分必要条件。他还研究了一些非线性差分方程，导出了由Neimark-Sacker分歧引起的不变曲线稳定性条件的计算公式，并给出了不变曲线的显式表达式。4）Kohda得到了半线性抛物方程初值问题解的爆破准则。这是描述通过使用一个超解和一个子解的固定问题。此外，他还给出了保证解爆破的条件。

The purpose of this project is to describe nonlinear phenomena mathematically by using asymptotic analysis. And we have the following results.1) Narukawa and Fukagai proposed a mathematical model related to the nonlinear elasticity. This is described by a degenerate quasilinear elliptic equation whose principal part has different orders at 0 and at infinity. They have showed a global bifurcation diagram of positive solutions for a nonlinear eigenvalue problem of such quasilinear equations and, in particular, the coexistence of multiple positive solutions. These results obtained by regularity estimate of weak solutions and modifying the argument given by Ambrosetti, Brezis and Cerami in the semilinear case.2) Ohnuma has investigated a class of singular degenerate parabolic equations including the p-Laplace diffusion equation and the equation of the mean curvature flow, and proved the comparison principle for these equations. He also discovered a strong maximum principle of quasilinear degenerate elliptic equations.3) Murakami showed a necessary and sufficient condition of the asymptotic stability of a fixed point for a higher order linear difference equation. He has also investigated some nonlinear difference equations, derived the formula to compute the stability conditions of the invariant curve caused by the Neimark-Sacker bifurcation and, moreover, given the explicit expression of the invariant curve.4) Kohda has obtained blow-up criteria for a solution of an initial value problem of a semilinear parabolic equation. This is described by using a super-solusion and a sub-solution of the stationary problem. Moreover, he had given some condition which guarantees the blow-up of the solution.